Differentiation matrices are used in spectral methods for the numerical solution of ordinary and partial differential equations, going back to their implicit use by Lanczos with the Chebyshev basisActually, Lanczos used the generalized inverse, \(\mathbf {D}_{\mathrm {Chebyshev}}^+\)  which turns out to be bidiagonal for the Chebyshev basis; this is simpler for hand computation. We will see this later in this section.. They can be used for quadrature, especially Filon or Levin quadrature, for highly oscillatory integrands. The first serious study seems to be [1]}. One of the present authors is working on this now [2]}. See also [3]}, [4]}, and Chapter 11 of [5]}.
